On free lunches in random walk markets with short-sale constraints and small transaction costs, and weak convergence to Gaussian continuous-time processes

2014, Vol. 28, No. 2, 223–240 DOI:10.1214/12-BJPS203

©Brazilian Statistical Association, 2014

On free lunches in random walk markets with short-sale constraints and small transaction costs, and weak convergence to Gaussian continuous-time processes

Nils Chr. Framstad^a,b

aUniversity of Oslo

bThe Financial Supervisory Authority of Norway

Abstract. This paper considers a sequence of discrete-time random walk markets with a safe and a single risky investment opportunity, and gives con- ditions for the existence of arbitrages or free lunches with vanishing risk, of the form of waiting to buy and selling the next period, with no shorting, and furthermore for weak convergence of the random walk to a Gaussian continuous-time stochastic process. The conditions are given in terms of the kernel representation with respect to ordinary Brownian motion and the dis- cretisation chosen. Arbitrage and free lunch with vanishing risk examples are established where the continuous-time analogue is arbitrage-free under small transaction costs—including for the semimartingale modifications of fractional Brownian motion suggested in the seminal Rogers [Math. Finance 7(1997) 95–105] article proving arbitrage in fBm models.

1 Introduction

If a continuous-time model for a financial market is discretised in time, will then the discretised version inherit its properties when it comes to free lunches, or ab- sence of such? Asking the converse question: if a continuous-time model is the weak limit—“weak” because this topology gives neighbouring profits/loss process distributions for a given strategy—of a sequence of discrete-time models, will free lunch properties or no free lunch properties carry over the limit transition?

There is actually no guarantee that this will be the case. In Shiryaev’s book (1999, Section VI.3), there are given stronger sufficient conditions for convergence to fair prices in terms of weak convergence of the(driving noise, pricing kernel) pair. This paper will show that if this joint convergence fails, then there is a wide range of problems where the arbitrage properties differ between the discretised prices and their weak limits, even when small transaction costs are introduced to the former.

This author’s initial interest in the problem at hand, emerges from a work by Sottinen (2001), who establishes a sequence of discrete-time binary symmetric random walk (semimartingale) markets, which (a) converges weakly to a Black–

Scholes market with prices being geometric fractional Brownian motion with

Key words and phrases.Stock price model, random walk, Gaussian processes, weak convergence.

Received June 2011; accepted June 2012.

223

Hurst parameter H >1/2, and (b) admits an arbitrage obtained by waiting for the right moment to buy (if nonnegative drift) or short sell (if nonpositive drift) the stock, and unwinding the position the very next period; the “right moment” is of course when you might with probability one know that the stock market beats the money market even if tomorrow is a bad day (in which case you buy), or waiting for the conversely adverse stock market (in which case you short-sell). Now frac- tional Brownian motion is not a semimartingale, and as is well known since Rogers (1997) (for the positively autocorrelated parameter range), it will introduce arbi- trages to canonical models where the ordinary Brownian motion does not. In view of this, there seems to have been a view that the result of Sottinen (2001) is due to specifics of the fBm, or at least its nonsemimartingale property, and this au- thor admits to having fallen prey to this interpretation, which—as we shall see—is inaccurate.

This paper sets out to show that the phenomenon discovered bySottinen(2001), is to be expected way more generally, including in the discretisation of arbitrage- free semimartingale price processes. As an example, we refer to Rogers(1997), who also proposes a parametrised semimartingale process whose moving average kernel converges to the fBm’s—preserving the long memory which was the reason for suggesting fBm as a driving noise in the first place, but eliminating the short memory which caused the arbitrage. It turns out that when attempting to discretise in a manner akin to the construction of Sottinen (2001), the long memory will introduce arbitrages, and the arbitrage property is robust enough to withstand even the introduction of a small transaction cost. It is then essential that the discretised version has bounded downside (cf. the results ofGuasoni(2006),Guasoni et al.

(2008)). A different example, admitting free lunch with vanishing risk (FLVR) in the discretisation, is the Ornstein–Uhlenbeck process.

We shall on one hand give sufficient conditions for the existence of arbitrage or FLVR of the form (i) wait for a possible time to buy, and then (ii) sell next period.

On the other, we give sufficient conditions for weak convergence of the discrete random walks to the Gaussian continuous-time counterparts. Because the results concerning arbitrages will requireboundedinnovations in the random walks, the weak convergence result (Theorem 2.2) will also be restricted to this case. Our main contributions compared to the previous literature (primarilySottinen(2001)), are summarised as follows:

• We cover a fairly general class of Gaussian processes, and give examples to the existence of arbitrages/FLVRs of the above-mentioned form.

• Furthermore, we point out that the arbitrage for discretised fBm can emerge from the (originally desirable) long-run memory of the process, even if the short-run memory (which causes the arbitrage in the continuous-time model) is modified as to obtain the semimartingale property, for example, as suggested byRogers(1997).

• We cover any negative drift term (a word which should be interpreted cautiously for nonsemimartingales) without shorting, as it turns out that the instantaneous growth from the noise term can tend to infinity.

• For the same reason the arbitrage may also admit sufficiently smalltransaction costs.

• We do not have to assume the discretised market to be binary (hence complete if arbitrage-free) with symmetric innovations. We will however assume bounded support, where the bound might depend on how fine the discretisation.

• Weak convergence of the driving noise is likewise shown in this more general setting.

2 The continuous-time and discrete-time market models

Our market has one “safe” asset, taken as numéraire and normalised to price=1, and one “risky” assetS⁽ⁿ⁾, which for eachnis a discretisation of a continuously evolving stochastic process S.S will be constructed from a drift processA with time-derivativea(t)and a driving noiseZ, assumed to be a Gaussian moving aver- age process with right-continuous sample paths and an adapted (hence upper limit of integration ist) kernel representation

Z(t)= ^t0

−∞K(t, s)dW (s)+ ^t

t0

K(t, s)dW (s)

(2.1)

=J (t)+ ^t

t₀ K(t, s)dW (s),

with respect to standard Brownian motionW, whereKis a given function satisfy- ing the following properties:

K is deterministic and piecewise-continuous, (2.2a)

K(t, s)=0 ifs≥t (2.2b)

and _t

−∞

K(t, s)²ds <∞ ∀t. (2.2c) We assume that the agent enters the market at a given timet0≥0.

Notice that (2.2c) follows from the assumed Gaussian distribution, as the ex- pression is in fact E[(Z(t))²]. Notice also that some representations involveKwith a definition split in order to achieve square integrability, compensating the distant past. A frequently occurring representation form (cf., e.g., Cheridito (2004)) is K(t, s)=κ(t−s)−κ(−s), and under the assumption that the kernel vanishes for s≥t, then we haveκ(−s)=0 fors≥0. We shall later give particular attention to such a form of the typeK(t, s)=κ((t−s)₊)fort > s > t0, where we not need to specify the definition belowt0.

We might choose to discretiseW on the entire time line; however,Jwill merely enter as a drift term, and we can equally well discretiseJ directly. We shall choose to do the latter. Hence, we start by discretising the time scale (equidistantly) in intervals of length 1/n, where for eachnwe define

s_i⁽ⁿ⁾=t0+i− nt0

n , (2.3)

where·is the floor function (rounding toward−∞). Then we discretiseW for t > t0 by replacing its normalised incrementsn^1/2· [W (s_i⁽ⁿ⁾₊₁)−W (s_i⁽ⁿ⁾)]by ran- dom variablesξ_i+1=ξ_i⁽ⁿ⁾₊₁. Now discretiseZ into

Z⁽ⁿ⁾(t)=J nt

n

+

nt−1 i=nt₀

K nt

n , s_i⁽ⁿ⁾

·n^−1/2ξ_i⁽ⁿ⁾₊₁. (2.4) ForAwe can takeA(t0)=0, as we are interested in increments only; we therefore defineAand its discretisation as

A(t)= ^t

t0

a(s)ds, A⁽ⁿ⁾(t)=1 n

nt−1 i=nt₀

as_i⁽ⁿ⁾ (2.5) and finally,Sand its discretisation are assumed, respectively defined, to satisfy

S=G(A+Z), S⁽ⁿ⁾=GA⁽ⁿ⁾+Z⁽ⁿ⁾. (2.6) The canonical choice isG to be the exponential function, but we shall not need this specific property; for Proposition3.2, we will however use convexity, and for Theorem2.2we shall need continuity. Except whenKvanishes, theS⁽ⁿ⁾and theξi

sequence will generate the same filtration, so the first of the following assumptions is not very restrictive:

Assumption/notation 2.1. We assume formulae (2.1) through (2.6) to hold, and furthermore:

• The filtration will be generated by the {ξi}, so that the information at timet, is generated by{ξi}i≤t n.

• By “step numberj,” we shall mean at times_j⁽ⁿ⁾. That means that the agent’s first chance of trading, is not at step 0, but at prices noted at stepj₀:= nt₀. Should this lead to a singularity due to for example, t0=0, K(t,0)= +∞, then we shall however eliminate this by assuming (without mention) thatt0 is>0 and irrational.

• We shall use the term “j-measurable” to mean measurable atstep numberj, that is, at time s_j⁽ⁿ⁾, and write Pj =P⁽ⁿ⁾_j for the probability measure conditional on the filtration generated up to this time/step and E_j =E⁽ⁿ⁾_j for the corresponding conditional expectation.

• The {ξ_i⁽ⁿ⁾}i,n will be mutually independent and each ξ_i⁽ⁿ⁾ bounded, and there exists some (common) constant ν >0 such that ess infξ_i⁽ⁿ⁾ <−ν and ess supξ_i⁽ⁿ⁾> ν.

• Sinceξ_j is independent of the past, we shall suppress the dependence of law in terms like e.g., ess sup_ξj which will denote the supremum over the (Pj-)essential support ofξ_j.

• Two pieces of notation:K₁ shall denote the partial derivative with respect to the first variable. The symbolshall mean “no smaller than and not a.s. equal.”

• ais assumed locally bounded, andGis assumed continuous and strictly increas- ing.

It should be remarked that it is unreasonable for an approximation to normalised standard Brownian motion that ν <1, but only in parts of Theorems3.7and3.8 shall we actually need that 0 is interior in the support. Bounded support will how- ever be essential for the arbitrage conditions, and the following result will be sim- plified by assuming a common bound:

Theorem 2.2 (Weak convergence). Suppose E[ξi] = 0, E[ξ_i²] = 1 and ess sup|ξi| ≤M <∞(all i,n).ThenZ⁽ⁿ⁾ converges weakly to Z, and for con- tinuousGalsoS⁽ⁿ⁾toS,on the Skorohod spaceD([t0, T]),everyT > t0.

Proof. The drift and the already occurred part will represent no issue, and we can takeA=A⁽ⁿ⁾=J =0 without loss of generality. Also, we can takeG to be the identity, as weak limits commute with continuous functionsG. Now convergence in finite-dimensional distributions follows like inSottinen(2001, Theorem 1): by the CLT, the limit is Gaussian with zero mean; for the covariances, the indepen- dence of theξi’s yields, forT ≥t≥t0

EZ⁽ⁿ⁾(T )Z⁽ⁿ⁾(t) =

nt−1 i=nt₀

K nT

n , s_i⁽ⁿ⁾

K nt

n , s_i⁽ⁿ⁾

·E ξ_i⁽ⁿ⁾₊₁

√n 2

(2.7) which is a Riemann sum converging to the desired value_t^t₀K(T , s)K(t, s)ds.

It remains to prove tightness, which will follow by a set of sufficient conditions given in Whitt (2007, Lemma 3.11(ii.b)). ForT ≥t+h≥t≥t0, we have

E_jZ⁽ⁿ⁾(t+h)−Z⁽ⁿ⁾(t)²

=

n(t+h)−1 i=nt₀

K

n(t+h) n , s_i⁽ⁿ⁾

−K nt

n , s_i⁽ⁿ⁾ 2

Ej ξ_i⁽ⁿ⁾₊₁

√n 2

≤M²

n sup

t∈(t₀,T−h)

n(t+h)−1 i=nt₀

K

n(t+h) n , s_i⁽ⁿ⁾

−K nt

n , s_i⁽ⁿ⁾ 2

.

For each n, lettn be an argsup. For any subsequence of tn converging to a limit pointt, we have convergence as a Riemann sum:¯

→M² _t¯+h

t₀

K(t¯+h, s)−K(t, s)¯ ²ds (2.8)

which by square integrability tends to 0 ashdoes.

For the discrete-time markets, we shall restrict ourselves to the following set of strategies:

Definition 2.3. Let n <∞ be given. For any natural q, a “q-period strategy”

(“single period strategy” ifq =1), consists of waiting until some stopping time t_∗=s_j⁽ⁿ⁾_∗ ≥s_j⁽ⁿ⁾

0 , buying aj_∗-measurable numberu >0 of units, holding these un- til a stopping timet^∗=s_j⁽ⁿ⁾∗ wherej^∗∈ {j_∗+1, . . . , j_∗+q}and then selling allu units.

The “net return” from this transaction is

R=Rj_∗,j^∗:=u·S⁽ⁿ⁾t^∗−S⁽ⁿ⁾(t_∗)−_∗+^∗·λ, (2.9) where λ_∗ and λ^∗ are the respective transaction costs for buying and selling, allowed to depend on prices and units like in Assumption2.4below.

The reason for the “λ” parameter is that we will consider the properties for small transaction costs, and it will be convenient to scale by a number. The main results will be carried out under for fixed transaction costs andu=1, and Proposition3.2 will show that this is sufficiently general. For the time being, assume the more parsimonious form for_∗,^∗:

Assumption 2.4. _∗=_∗(u, S⁽ⁿ⁾(t_∗)) and ^∗=^∗(u, S⁽ⁿ⁾(t_∗), S⁽ⁿ⁾(t^∗)) will be nonnegative functions, bounded in (S⁽ⁿ⁾(t_∗), S⁽ⁿ⁾(t^∗)), whileλwill be a number

≥0.

Definition 2.5. We shall use the term “transaction costλ” to imply thatu=1 and _∗+^∗=1 (identically), and we shall refer to “the simple model (2.10)” the single period case of transaction costλwhereGis the identity.

This “simple model” will be the main focus. For this case, the net return on the event{j_∗<∞}will be

S⁽ⁿ⁾

t0+j_∗+1− nt0 n

−S⁽ⁿ⁾

t0+j_∗− nt0 n

−λ

(2.10a)

=x_j⁽ⁿ⁾_∗ +y_j⁽ⁿ⁾_∗ +z⁽ⁿ⁾_j∗+1−λ,

where we introduce the notation xj =x_j⁽ⁿ⁾= 1

n

as_j⁽ⁿ⁾+Js_j⁽ⁿ⁾₊₁−Js_j⁽ⁿ⁾, (2.10b)

yj =y_j⁽ⁿ⁾= 1

√n

j−1 i=j0

Ks_j⁽ⁿ⁾₊₁, s_i⁽ⁿ⁾−Ks_j⁽ⁿ⁾, s_i⁽ⁿ⁾ ξ_i⁽ⁿ⁾₊₁, (2.10c)

zj+1=z⁽ⁿ⁾_j+1= 1

√nKs_j⁽ⁿ⁾₊₁, s_j⁽ⁿ⁾ξ_j⁽ⁿ⁾₊₁ (2.10d) adopting the convention that the empty sum, corresponding toj=j₀(= nt₀), is zero. Notice that thez⁽ⁿ⁾_j+1term will represent theinnovationfrom stepj toj+1, and has subscript “j+1” since it is onlyj+1-measurable. Thexj andyj, which correspond to the memory of the process as well as the drifta, arej-measurable.

An arbitrage will occur in the market if the memory contribution dominates even in the worst-case innovation. The next section will make this more precise.

3 Free lunches: Sufficient conditions

Starting out with the definitions from the previous section, we now define arbi- trage and free lunches with vanishing risk under our admissibility conditions. In- formally, we have a FLVR if we can obtain an arbitrarily small downside to mean return ratio, and an arbitrage if one can have positive-mean return without down- side. The arguably most natural, and also the strictest, concept of “downside” is the worst-case outcome, the essential supremum of the negative part, and we shall restrict ourselves to such a definition. The following definition appears notation- ally a bit cryptic, but will underq-period strategies coincide with the conventional definition of FLVR and arbitrage; informally, it says that downside should be arbi- trarily small compared to expected return (and we note that the expectation term is finite, by boundedness of theξi). Note however that this diverges on the outset—

though not in substance, as we shall see below—from a commonplace assumption of fixed horizon. This fixed horizon is less natural here, where the strategies in- volve waiting first and only then is there a bound on the holding period.

Definition 3.1. Fixn, qboth<∞. Consider the condition ess inf^(Pj∗)[R_j∗,j∗|D_j∗]

E[Rj_∗,j^∗|Dj] >−δ and E[Rj_∗,j^∗|Dj_∗] ∈(0,∞). (3.1)

• The market is said to admitfree lunch with vanishing risk(“FLVR”)inqperiods if for everyδ >0 there exist stopping timesj_∗≥j0andj^∗∈ {j_∗+1, . . . , j_∗+q} and aj_∗-measurable eventD_j∗ with P_j0[D_j∗]>0, such that (3.1) holds.

• The FLVR is called anarbitrageif the FLVR definition holds also forδ=0.

• The simple model (2.10) will be said to admit FLVR, respectively, arbitrage, if the respective definition applies withq=1 (i.e.,j^∗=j_∗+1).

Whenever necessary to distinguish the ex ante (atj0) random variable which is ei- ther 0 (ifDj does not occur) or>0 on one hand, from theDj-conditional positive return on the other—colloquially speaking, the lottery ticket that yields0 from the actual positive lunch prize—we shall use terms like the “event” that the lunch

“manifests itself.”

Obviously, the lack of time bound makes no difference for an arbitrage; if there is an arbitrage according to this definition, then for some fixedQ, there is an ar- bitrage which is closed out within Q steps, that is,j^∗≤Q. Conversely, it does not matter thatq is assumed deterministic; had we employed the same definition except withqbeing merely measurable and finite, we would have had an arbitrage for some deterministicq as well. The FLVR definition, on the other hand, might require an unboundedj_∗. Informally, a FLVR is a sequence of lunches with uni- formly positive mean, but where the risk tends to zero. This means that for any nonzero downside you choose as tolerance, then there is a fixedQsuch that you have a lunch within your risk tolerance withinQsteps. Letting downside tend to zero, then our setup allowsQ to grow, as long as q obeys a fixed bound; com- pare this to the usual Black–Scholes setup, which rules out the strategy of trading and waiting for the unbounded stopping time until your position has made a given profit.

For the purpose of givingsufficientconditions for arbitrage/FLVR undersmall transaction cost—which is the main object of this section—the simple model (2.10), for whichq=1, turns out fairly close to general.

Proposition 3.2 (Free lunches in the simple model (2.10) vs. in the full model).

Fixu >0, n <∞.Assume that^∗of at most linear growth w.r.t.the last variable (the selling price).Then there is an arbitrage for sufficiently smallλ,provided that so is the case in the simple model(2.10).If G is convex, then there is FLVR for sufficiently smallλ,provided that so is the case in the simple model(2.10).

Proof. The proof is less interesting, and is relegated to theAppendix. Notice that if ξjK(s_j⁽ⁿ⁾₊₁, s_j⁽ⁿ⁾) is upper bounded (for every j and n), then the at most linear growth condition will hold (since an arbitrage must be closed out in a bounded

number of periods).

Informally, an arbitrage in the simple model (2.10), occurs if at some bounded j_∗∈ [j0,∞), given the information available then, the transaction costs plus the worst-case possible downside from the innovation z⁽ⁿ⁾_j

∗+1 will be more than fully compensated by the contribution from the dependence of the past (i.e., x_j⁽ⁿ⁾_∗+1 +y_j⁽ⁿ⁾_∗+1); a FLVR occurs if it is “sufficiently more than compensated in

mean” and “nearly fully compensated in worst-case.” The following result is key for the arbitrage case.

Proposition 3.3 (Sufficient conditions for arbitrage in the simple model (2.10)).

Fixn <∞.If for some naturalj≥j0,we have ess sup

{ξ_i}i=j0+1,...,j

{x_j +y_j} +ess inf

ξ_j+1 z_j+1≥ ¯λ≥0 (3.2) we have an arbitrage for all transaction costsλ∈ [0,λ)¯ by choosing j_∗=thisj. Furthermore,we have arbitrage for transaction costλ¯ if in addition there is a point probability thatzj+1attains itsess sup.

Proof. Suppose (3.2) holds for someλ. Let¯ Dj be thej-measurable event of at- taining

x_j +y_j ≥ ess sup

{ξi}_i=j0+1,...,j{x_j +y_j} −ε.

Then Pj₀[Dj]>0 for eachε >0; letε∈(0,λ¯−λ)if nonempty. Should the event Dj occur at step j, then (2.10a) is>0 and so is the ess inf of (3.1), where then both numerator and denominator become positive. Arbitrage also for transaction costλ¯ holds if we have positive probability atε=0.

So the discrete market will admit an arbitrage if there may occur a period so good that the contribution from this beneficial history, knocks out the innovation so much that the worst-case scenario is a profit. Evidently, this will not happen if Kis a constant (i.e., ordinary Brownian motion), for then the history does not mat- ter; on the other hand, ifK is increasing in its first variable, then it is near-trivial to construct arbitrage examples by letting downside be bounded and the upside be unbounded. While one can certainly imagine a some modeler trying to use, for example, a shifted lognormal in order to model limited liability investments, a choice of a symmetric distribution for the ξi would arguably be more natural and innocuous-looking. But for a suitably wide range of models, a larger down- side than upside will not even prevent an arbitrage, as we shall soon see. For the book-keeping of “good” and “bad” ξi outcomes, we shall denote their essential suprema/infima as follows:

M_i=M_i⁽ⁿ⁾=ess supξ_i⁽ⁿ⁾, (3.3a) mi=m⁽ⁿ⁾_i = −ess infξ_i⁽ⁿ⁾. (3.3b) Standing at step j, then the worst that can happen in the next innovation, cf.

(2.10c), is denotedβ_j (“beta” for “bad”):

β_j =β_j⁽ⁿ⁾=

−m_j ifKs_j⁽ⁿ⁾₊₁, s_j⁽ⁿ⁾≥0,

Mj otherwise. (3.4a)

Looking back in time, we defineγij (“gamma” for “good” ) to be the best possible history overi=j0, . . . , j (cf. (2.10b)):

γij=γ_ij⁽ⁿ⁾=

Mi ifKs_j⁽ⁿ⁾₊₁, s_i⁽ⁿ⁾≥Ks_j⁽ⁿ⁾, s_i⁽ⁿ⁾,

−mi otherwise. (3.4b)

We want to specify this in terms of time, not only steps. Suppose we are targeting an arbitrage within timeT for the discretised model, choosingj_∗to be the second- to-last step before timeT, closing out the transaction before timeT:

j_∗+1=nT −nt0+ nt0. (3.5) Then define (T , s)=⁽ⁿ⁾(T , s) as a left-continuous step function with values (T , s_i⁽ⁿ⁾)=γ_ij∗; extend it to be 0 for s∈ [s_j⁽ⁿ⁾_∗ , T]. Consider then n^−1/2y_j∗ and bear in mind thatj_∗depends onnchosen as (3.5). Then, provided that limits exist (again,K₁ denotes the derivative w.r.t. the first variable), we have

limn (ess supyj_∗

√n)≥ ^T

t0

K₁(T , s)lim inf

n ⁽ⁿ⁾(T , s)ds,

where the inequality follows from nonnegativity of the integrand and the Fatou lemma. So, givenε >0, then for all large enough but finiten, there will be positive P_j0measure of the event

Dj_∗=

yj_∗≥ ^T

t0

K₁(T , s)⁽ⁿ⁾(T , s)ds−ε

n^−1/2

. (3.6)

This gives rise to the following result.

Theorem 3.4 (Sufficient conditions for arbitrage within timeT in the simple model (2.10)). Fix aT > t0 and for eachn,letj_∗be given by(3.5).Assume that atT we haveJ Hölder continuous with exponentα >1/2,and furthermore that K(t, s)is differentiable in the first variable,att=T,for eachs∈(t0, T ).Then if

_T

t₀ K₁(T , s)⁽ⁿ⁾(T , s)ds >Ks_j⁽ⁿ⁾_∗+1, s_j⁽ⁿ⁾_∗ β_j⁽ⁿ⁾+ ¯ε (3.7) holds for all large enoughn,someε >¯ 0,then for anynlarge enough,there is an arbitrage with sufficiently small transaction costs,by waiting until stepj_∗. Proof. Under the assumptions, we would on the event in (3.6) have a net return of at least

a(s_j⁽ⁿ⁾_∗ )+J (s_j⁽ⁿ⁾_∗+1)−J (s_j⁽ⁿ⁾_∗ )

n^1/2 + ^T

t₀

K₁(T , s)⁽ⁿ⁾(T , s)ds

−Ks_j⁽ⁿ⁾_∗+1, s_j⁽ⁿ⁾_∗ β_j⁽ⁿ⁾−ε

n^−1/2−λ

and the Hölder regularity ensures that the first term inside the bracket (i.e.,xj√ n), will vanish asngrows. Then by (3.7), the net return will for large enoughnbe a positive random variable, even with small transaction costs.

Remark 3.5. First, observe that if lim_sTK(T , s)=0, then this would lead to arbitrages. Second, note that Theorem3.4is stated for fixedT, but it is sufficient to look for someT where it applies. For example, ifξihave symmetric support for eachn, then we can replace (3.7) by

sup _T

t₀

K₁(T , s)ds−K

T , T − 1 n

>0, (3.8)

where the sup is taken over thoseT > t0 for whichn(T −t0)is integer. Now one can look for arbitrages by lettingT grow.

Theorem 3.4also applies to semimartingales. The corollary is stated only for the natural choice of symmetric innovations.

Corollary 3.6. There are infinite-variation semimartingales Z, equalling weak limits of their discretisationsZ⁽ⁿ⁾formed by i.i.d.bounded symmetricξi,for which Theorem3.4applies.

Proof. Put t₀=0 for simplicity. From Cheridito (2004, Theorem 3.9), it is suf- ficient for the semimartingale property thatK(t, s)=κ(t −s)ont > s >0 with κ being continuous and piecewise differentiable withκ∈L²((0,∞)), and under these conditions, total variation is infinite on compacts iff κ(0⁺)=0. Choose a κ≥0 with a global maximum atT, withκ(T ) >2κ(0⁺) >0; then it satisfies the hypothesis of Theorem3.4, and we only need κ(ϑ)to be smooth and κ to tend

sufficiently fast to 0 as to be square integrable.

The form where the dependence on (t, s) only appear through the difference, will cover many cases and simplify calculations. We introduce the conditions:

For eachn, we havei-independentmi=m=m⁽ⁿ⁾

andM_i=M=M⁽ⁿ⁾, (3.9a)

K(t, s)=κ(t−s)₊ fors≥t0, withκ not constant on(0,∞) (3.9b)

—the nonconstantness ruling out the ordinary Brownian motion. As seen above, this form covers a wide class of even semimartingales. We can then writeyandz as

yj = 1

√n

j−1 i=j₀

κ

j+1−i n

−κ j−i

n

ξ_i⁽ⁿ⁾₊₁, (3.10a)

zj+1= 1

√nκ 1

n

ξ_j⁽ⁿ⁾₊₁. (3.10b)

Now consider the good outcomesγij; if κ is monotone orm=M, then the se- ries in (3.10a) will telescope. A nonmonotoneκ only has more variation, which increases the sum, so the ess sup ofy_j will therefore be at least

κ

j−j₀+1 n

−κ 1

n √

n

(3.11)

·

⎧⎨

⎩M⁽ⁿ⁾ ifκ

j−j0+1 n

≥κ 1

n

, m⁽ⁿ⁾ otherwise

(if we want to utilise the variation of κ, we could write in terms as a sum of

|κ|-terms, tending to the constant times the total variation ofκ over the interval (1/n, (j−j0+1)/n)). We have the following theorem.

Theorem 3.7 (Sufficient conditions for arbitrage in the simple model under the form (3.9)). Suppose thatJ is Hölder continuous with indexα >1/2and fur- thermore that for some subsequencen,we havem⁽ⁿ⁾,andM⁽ⁿ⁾bounded.Then, each of the following conditions implies arbitrage for all large enough —and furthermore,for each of thosen,the arbitrage admits small enough transaction costs:

(a) lim inf|κ(1/n)| =0,orκchanges sign.

(b) The total variation ofκ over(0,∞)(i.e.,₀^∞|κ(ϑ)|dϑ ifκexists),is>than lim inf

κ(1/n)·max{M⁽ⁿ⁾, m⁽ⁿ⁾} min{M⁽ⁿ⁾, m⁽ⁿ⁾}

.

Proof. In all cases, Hölder regularity ensures thatxj√

nwill tend to 0 asngrows.

Then:

(a) By (3.9b),κ takes some nonzero value. Suppose first thatκ(1/n)0 while κ(ϑ) >0. Then choosing a sequence of j’s so that(j −j0+1)/napproxi- mates ϑ from the appropriate side (recall that (2.2a) assumes only piecewise continuity), we obtain

√n· [zj+1+ess supyj+1] ≥ M·

κ

j−j0+1 n

−κ 1

n

− m Mκ

1 n

→Mκ(ϑ) >0.

The negative-sign case likewise converges to m· |κ(ϑ)|. Now suppose that κ changes sign, and by the previous part of the proof, we can assume that κ(1/n) is bounded away from 0. Suppose that κ(1/n) >0> κ(ϑ). Then choosingj as above, we obtain

√n· [zj+1+ess supyj+1] ≥m·

κ 1

n

−κ

j−j₀+1 n

−κ 1

n

= −mκ

j−j0+1 n

which is positive whenever nis large enough. The case with reversed signs follows likewise.

(b) √

ness supyj exceeds min{m⁽ⁿ⁾, M⁽ⁿ⁾} ×total variation on(_n¹,^j−j_n⁰⁺¹), while even in the worst case,√

nzj ≥ −|κ(1/n)| ·max{m⁽ⁿ⁾, M⁽ⁿ⁾}. The Hölder regularity condition onJ in Theorems3.4and3.7admits ramifica- tions, as we need only bound the downside—it can be replaced by the condition that for eachnwe havexj ≥0 for infinitely manyj, which by symmetry ofW oc- curs in at least half of the cases (unconditionally, i.e., “P_−∞”). Shouldxj√

nblow up as n grows, then it would be expected thatJ (T , T −1/n)oscillates around 0, and we would be able to extract a subsequence where it adds positively to the return. Similar considerations would improve upon the next Theorem3.8as well.

Before we state that result, it should be noted that the total variation criterion can also be improved upon in the setup of Theorem3.4, if the variation of the step func- tion corresponding to grid size 1/n, diverges as time grows. When|κ(1/n)| → ∞, andκ is monotone (anything else improves total variation) and does not change sign (if it does, Theorem3.4part (a) applies), we can still have free lunches with vanishing risk.

Theorem 3.8 (Sufficient conditions for FLVR under the form (3.9)). Sup- pose zero transaction cost and that M⁽ⁿ⁾ =m⁽ⁿ⁾ and infiE[ξ_i⁽ⁿ⁾]/m⁽ⁿ⁾ >−1.

Furthermore, assume that κ does not change sign, and that infϑ|κ(ϑ)| =0<

lim infn|κ(1/n)| (possibly= +∞). Then either of the following is sufficient for FLVR:

(a) For givenn:x_j⁽ⁿ⁾≥0for infinitely manyj,and additionally, lim_ϑ→∞κ(ϑ)= 0.

(b) J is Hölder continuous with index >1/2, lim infn

inf_iE[ξ_i⁽ⁿ⁾]

m⁽ⁿ⁾ >−1, andn is large enough.The FLVR is an arbitrage if the positivity of thexj is uniform (w.r.t.j).

Proof. We prove only the case of positiveκ. Just like in the proof of Theorem3.7 part (a), √

n(ess supyj +ess infzj+1) will telescope to −m⁽ⁿ⁾κ(^j−j_n⁰⁺¹), which can be made arbitrarily close to 0; a Pj₀-positive event will be it falling withinε/n of−m⁽ⁿ⁾κ(^j−j_n⁰⁺¹). It will turn out that this takes care of the numerator of (3.1) in the FLVR definition. For the denominator, we need the expected return:

x_j +ess supy_j+E[z_j+1]

=x_j + 1

√n

−mκ

j−j0+1 n

+mκ

1 n

+κ

1 n

Eξ_j+1

(3.12)

=xj + m

√n

−κ

j−j0+1 n

+κ

1 n

1+Eξj+1

m

.

(a) Now passing through a subsequence with nonnegativexj, thenκ(^j−j_n⁰⁺¹)will vanish and (3.12) will be positive from the assumption; this takes care of the denominator of (3.1). Assuming that the numerator is negative (otherwise there is arbitrage), the ratio (3.1) will on the Pj₀-positive event of the history falling withinε/√

nof its ess sup, exceed

−mκ

j−j0+1 n

−εm

−κ

j−j0+1 n

+κ

1 n

1+Eξj+1

m which tends to 0 asj andε⁻¹grow.

(b) The ratio (3.1) becomes—withε=ε_npossiblyn-dependent—

m

√n xj√

n m −κ

j−j0+1 n

− ε m√

n

m

√n xj

√n m −κ

j−j0+1 n

− ε m√

n+κ 1

n

1+Eξj+1

m

.

First, cancelm/√

n. Then, observe that as in the proof of Theorem3.7part (a), we can choose j depending on n as to approximate the appropriate ϑ, or possibly the appropriate sequence of ϑ’s, so that κ(^j−j_n⁰⁺¹) vanishes in the limit, along with—by assumption—everything involvingxj. Then for suitably smallε, thenκ(1/n)will make the denominator (and hence the expectation) positive, while the numerator can be chosen arbitrarily small.

Remark 3.9. Notice that the statements of Theorem 3.7 and of Theorem 3.8 part (b), do not depend on whatt0is, and what history the agent faces upon entry.

It is certainly not obvious that it should be this way. The setup of Theorem3.4does not rule out a priori that there could be an arbitrage initially, to be exploited at a later stage if a positive eventDj occurs, but which with positive probability dis- appears for good. (In more formal terms, theξj+1could be drawn so that not only would Pj₀+1[Dj] =0, but also Pj₀+1[Pi[Dj]>0] =0 for alli > j0.) But under the applicability of Theorem3.7or Theorem3.8part (b), this will not be the case:

the arbitrage, respectively, FLVR, will show up for large enoughn regardless of whether the agent enters after a long “bad” period which hampers future prospects;

for fine enough discretisation, there will always be a positive event where the ar- bitrage could materialise. This is not to say that the probability of this event is independent of history, nor that the choice of positive event is—only the binary question of existence. Under these results, regardless how disadvantageous the de- velopment has been, the strategy of calmly waiting for lunch time, will always yield positive expected value.

4 Some examples and nonexamples

We will in the following discuss a few cases. Throughout this section, assume common symmetric support[−m, m].

(a) Brownian motion is not prone to arbitrages in the discretised version; we have K(t, s)=1t >s>0, so there is no contribution from history.

(b) The Ornstein–Uhlenbeck process (mean-reverting to 0) admits the representa- tionκ(ϑ)=κ(0)·e^−vt with v >0. Thisκ satisfies Theorem3.8, which will yield FLVR (but, easily, not arbitrage) if choosing the distribution of the ξ_i to comply with the assumptions. If there is positive drift (mean-reversion to a positive level μ), then the discretised version admits arbitrage. It should be noted though, that in the continuous-time model, a portfolio of η(t) yields a wealth process dynamics of η(t)dZ(t)=η(t)[v·(μ−Z(t))+σdW (t)], where there is an arbitrarily big upside for given volatility level, by waiting for Z to become negatively large. However, the continuous model remains arbitrage-free, regardless ofμ.

(c) Sottinen (2001) considers fractional Brownian motion with Hurst parameter H >1/2, using the representation

K(t, s)= ^t

s

(u/s)^H−1/2(u−s)^H−3/2du

(up to an irrelevant positive constant), so that K(t⁺, t)=0 and K₁ is pos- itive. Then by Remark 3.5 we will have Theorem 3.4 applying, as J (t)= t₀

0 _t

s(u/s)^H−1/2(u−s)^H−3/2dudW (s)is differentiable at t=T⁻ (just in- terchange order of integration).

Furthermore, by Remark3.9, the arbitrage holds regardless of history. No matter how bad (and how long!) the initial period until entry is, there is still a positive event that a free lunch will actually manifest.

(d) Maybe a more common representation for fractional Brownian motion is, for anyH=1/2 and up to a constant,

K(t, s)=(t−s)^H−1/2−(−s)₊^H−1/2,

corresponding to κ(ϑ)=ϑ^H−1/2. Let us assume that the ξi have the same support. κ is monotonous, so then conditions (3.9) hold. Now the results are different for positively (H >1/2) and negatively (H <1/2) autocorrelated fBm:

• In the case H > 1/2, κ(0⁺)=0 and κ is increasing. Furthermore, J is Hölder continuous of order up toH. Theorem3.4applies, by Remark3.5.

• In the caseH <1/2,κ(0⁺)= ∞whileκ(∞)=0. Then for at least half of the cases, Theorem3.8part (a) applies.

(e) Rogers(1997) proposes a modification of fractional Brownian motion, in order to eliminate the arbitrage but preserve the long run memory properties which motivated the use of fBm in finance in the first place. Rogers gives a specific (monotone) example

κ(ϑ)=k·ϑ²+v^(H−1/2)/2, (4.1)

but suggests more generally to choose κ such that κ(0)=1, κ(0)=1, and has the same∼ϑ^H−1/2 behaviour for largeϑ. This behaviour, tending to∞ for H >1/2 and 0 forH <1/2, is sufficient to yield the same results as for example (d).

(f) For a mix between a fractional and an (uncorrelated) ordinary Brownian mo- tion, there is a very peculiar result byCheridito (2001): if the fBm part has H >3/4, then it behaves just as drift (which, e.g., means that it does not enter in the Black–Scholes formula). ForH≤3/4, there is still arbitrage as if there were no Brownian component. However, in our case, such a process works like example (e) when H >1/2: mixing in Brownian motion at volatility σ, we getκ(t−s)replaced byσ²+κ(t−s)²=(σ²+(t−s)^2H−1)^1/2 which for H >1/2 works analogous to (4.1), and will admit arbitrage in the dis- cretisation. There is apparently nothing happening at the Cheridito threshold of 3/4.

We end this section by pointing out that not only are the discrete markets possi- bly different than their weak limits when it comes toexistenceof free lunches; the arbitrages themselves might occur from different properties ofK. In the canonical models, either of the properties κ(0⁺)=0 and κ(+∞)= +∞ will lead to arbi- trages, and the latter will be due to the long-run memory. The long-run memory was arguably the reason why fractional Brownian motion κ(ϑ)=ϑ^H−1/2 (with H >1/2) was suggested in the first place as driving noise for financial mar- kets, and the Rogers proposal of example (e) above, leaves that long memory in the process. Let assume that t0 =0, and that, coincidentally,x_j0=x_j0+1 =0 in order to isolate short-term effects. Fix n for the moment. Then the first-step innovation is symmetric, and the next one cannot lead to arbitrage either, as κ(2/n)−κ(1/n) < κ(1/n)by concavity. The minimum number of steps (afterj0) for the arbitrage for the fBm case, is the smallest integer>2^1/(H−1/2) (equality suffices if the ess sup has point mass), so that the absolute minimum is 5 steps, obtained forH above≈0.931. Of course, as the partition refines andngrows, this happens in shorter time, thus approaching the continuous-time setup where the profit instantly increases from 0 (Framstad(2004)). On the other hand, even when mixed with ordinary Brownian motion, the Examples (e) and (f) yield arbitrages;

boosting up the ordinary Brownian part, will merely require a longer beneficial period before the arbitrage manifests itself. Those arbitrages are due to the long runbehaviour—namely, the fact thatκtends to+∞on the long run.

5 Concluding remarks

We have seen that discrete-time random walk markets may behave radically dif- ferent from their weak limits, as the former may admit arbitrages or FLVRs which vanish in the limit. Furthermore, quite a few of our estimates may be sharpened